Approximation in the mean by rational functions (1904.06446v6)

Abstract: For $1\le t < \infty$, a compact subset $K\subset\mathbb C$, and a finite positive measure $\mu$ supported on $K$, $R^t(K, \mu)$ denotes the closure in $L^t(\mu)$ of rational functions with poles off $K$. Let $\text{abpe}(R^t(K, \mu))$ denote the set of analytic bounded point evaluations. The objective of this paper is to describe the structure of $R^t(K, \mu)$. In the work of Thomson on describing the closure in $L^t(\mu)$ of analytic polynomials, $P^t(\mu)$, the existence of analytic bounded point evaluations plays critical roles, while $\text{abpe}(R^t(K, \mu))$ may be empty. We introduce the concept of non-removable boundary $\mathcal F$ such that the removable set $\mathcal R = K\setminus \mathcal F$ contains $\text{abpe}(R^t(K, \mu))$. Recent remarkable developments in analytic capacity and Cauchy transform provide us the necessary tools to describe $\mathcal F$ and obtain structural results for $R^t(K, \mu)$. Assume that $R^t(K, \mu)$ does not have a direct $L^t$ summand. Let $H^\infty_{\mathcal R}(\mathcal L^2_{\mathcal R})$ be the weak$^*$ closure in $L^\infty (\mathcal L^2_{\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\mathcal F$, where $\mathcal L^2_{\mathcal R}$ denotes the planar Lebesgue measure restricted to $\mathcal R$. We prove that the identity map ($r\rightarrow r$, $r$ is a rational function with poles off $K$) extends an isometric isomorphism and a weak$^*$ homeomorphism from $R^t(K, \mu)\cap L^\infty(\mu )$ onto $H^\infty_{\mathcal R}(\mathcal L^2_{\mathcal R })$. Consequently, we show that a decomposition theorem (Main Theorem II) of $R^t(K, \mu)$ holds for an arbitrary compact subset $K$ and a finite positive measure $\mu$ supported on $K$, which extends the central results regarding $P^t(\mu)$.